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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Particular solutions of a certain class of associated Cauchy-Euler fractional partial differential equations via fractional calculus

Shy-Der Lin*, Chia-Hung Lu and Shou-Mei Su

Author Affiliations

Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, 32023, Taiwan, R.O.C

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Boundary Value Problems 2013, 2013:126  doi:10.1186/1687-2770-2013-126


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/126


Received:30 January 2013
Accepted:1 May 2013
Published:16 May 2013

© 2013 Lin et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields of science and engineering. Many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The purpose of this paper is to present a certain class of the explicit particular solutions of the associated Cauchy-Euler fractional partial differential equation of arbitrary real or complex orders and their applications as follows:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2">View MathML</a>; A, B, C, M, N, α and β are arbitrary constants.

MSC: 26A33, 33C10, 34A05.

Keywords:
fractional calculus; differential equation; partial differential equation; generalized Leibniz rule; analytic function; index law; linearity property; principle value; initial and boundary value

Dedication

Dedicated to Professor Hari M Srivastava.

1 Introduction, definitions and preliminaries

The subject of fractional calculus (that is, derivatives and integrals of any real or complex order) has gained importance and popularity during the past two decades or so, due mainly to its demonstrated applications in numerous seemingly diverse fields of science and engineering (cf.[1-15]). By applying the following definition of a fractional differential (that is, fractional derivative and fractional integral) of order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M3">View MathML</a>, many authors have obtained particular solutions of a number of families of homogeneous (as well as nonhomogeneous) linear fractional differ-integral equations.

In this paper, we present a direct way to obtain explicit solutions of such types of the associated Cauchy-Euler fractional partial differential equation with initial and boundary values. The results are a coincidence that the solutions are obtained by the methods applying the Laplace transform with the residue theorem. In this paper, we present some useful definitions and preliminaries for the paper as follows.

Definitions 1.1 (cf.[6-10])

If the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4">View MathML</a> is analytic and has no branch point inside and on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M5">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M6">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M7">View MathML</a> is an integral curve along the cut joining the points z and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M9">View MathML</a> is an integral curve along the cut joining the points z and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M10">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M11">View MathML</a>

(1.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M12">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M13">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M14">View MathML</a>

(1.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M15">View MathML</a>

(1.5)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M17">View MathML</a>) is said to be the fractional derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4">View MathML</a> of order ν and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M16">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M20">View MathML</a>) is said to be the fractional integral of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4">View MathML</a> of order −ν, provided that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M22">View MathML</a>

(1.6)

First of all, we find it is worthwhile to recall here the following useful lemmas and properties associated with the fractional differ-integration which is defined above.

Lemma 1.1 (Linearity property)

If the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M24">View MathML</a>are single-valued and analytic in some domain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M25">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M26">View MathML</a>

(1.7)

for any constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M27">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M28">View MathML</a>.

Lemma 1.2 (Index law)

If the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4">View MathML</a>is single-valued and analytic in some domain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M25">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M31">View MathML</a>

(1.8)

Lemma 1.3 (Generalized Leibniz rule)

If the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M4">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M24">View MathML</a>are single-valued and analytic in some domain<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M25">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M35">View MathML</a>

(1.9)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M36">View MathML</a>is the ordinary derivative of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M24">View MathML</a>of ordern (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M38">View MathML</a>), it being tacitly assumed (for simplicity) that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M24">View MathML</a>is the polynomial part (if any) of the product<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M40">View MathML</a>.

Lemma 1.4 (Cauchy’s residue theorem)

Let Ω be a simple connected domain, and letCbe a simple closed positively oriented contour that lies in Ω. Iffis analytic insideCand on C, expect at the point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M41">View MathML</a>that lie insideC, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M42">View MathML</a>

(I) Iffhas a simple pole at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M43">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M44">View MathML</a>

(II) Iffhas a pole of orderkat<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M43">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M46">View MathML</a>

Property 1.1 (cf.[6-10])

For a constanta,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M47">View MathML</a>

(1.10)

Proof The proofs between ‘ν is not an integer’ and ‘ν is an integer’ are not coincident, so we mention the proof as follows.

In case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M48">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M49">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M50">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M51">View MathML</a>.

In case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M52">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M53">View MathML</a>

Therefore we have Property 1.1 for arbitrary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M54">View MathML</a>. □

Property 1.2For a constanta,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M55">View MathML</a>

(1.11)

Property 1.3For a constanta,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M56">View MathML</a>

(1.12)

Property 1.4 (cf.[2,16,17])

The fractional derivative of a causal function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M57">View MathML</a>is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M58">View MathML</a>

(1.13)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M59">View MathML</a>denotes the ordinary derivative of ordernand Γ is the gamma function.

The Laplace transform of a function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M57">View MathML</a>is denoted as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M61">View MathML</a>

(1.14)

wheresis the Laplace complex parameter. We recall from the fundamental formula (cf[16])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M62">View MathML</a>

(1.15)

2 Main results

Theorem 2.1The fractional partial differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M63">View MathML</a>

(2.1)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M66">View MathML</a>andA (≠0), B, C, M, Nare constants, has its solutions of the form given by

(a)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M67">View MathML</a>

(2.2)

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M68">View MathML</a>;

(b)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M69">View MathML</a>

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M70">View MathML</a>, and the roots<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72">View MathML</a>of Equation (2.5) are repeated; that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73">View MathML</a>;

(c)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M74">View MathML</a>

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M75">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M77">View MathML</a>are the conjugate pair roots of Equation (2.5).

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M78">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M79">View MathML</a>

(2.3)

So that the given equation (2.1) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M80">View MathML</a>

(2.4)

Equation (2.4) leads to the auxiliary equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M81">View MathML</a>

(2.5)

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M82">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M83">View MathML</a>

are the two roots of the auxiliary equation (2.5). Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M84">View MathML</a> is a solution of the fractional partial differential equation (2.1) whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M85">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M86">View MathML</a>) is a solution of the auxiliary equation (2.5).

There are three different cases to be considered, depending on whether the roots of this quadratic equation (2.5) are distinct real roots, equal real roots (repeated real roots), or complex roots (roots appear as a conjugate pair). The three cases are due to the discriminant of the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M87">View MathML</a>.

• Case I: Distinct real roots (when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M88">View MathML</a>).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72">View MathML</a> denote the real roots of Equation (2.5) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M91">View MathML</a>. Then the general solution of Equation (2.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M92">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M86">View MathML</a>) are constants.

• Case II: Repeated real roots (when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M95">View MathML</a>).

If the roots of Equation (2.5) are repeated, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73">View MathML</a>, then the general solution of Equation (2.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M97">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M99">View MathML</a>) are constants.

• Case III: Conjugate complex roots (when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M100">View MathML</a>).

If the roots of Equation (2.5) are the conjugate pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M102">View MathML</a>, then a solution of Equation (2.1) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M103">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M99">View MathML</a>) are constants.

In general,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M106">View MathML</a>

forms a fundamental solution, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M28">View MathML</a> and λ are constants. □

Theorem 2.2The fractional partial differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M109">View MathML</a>

(2.6)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M66">View MathML</a>andA (≠0), B, C, M, Nare constants, has its solutions of the form given by

(a′)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M113">View MathML</a>

(2.7)

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M114">View MathML</a>;

(b′)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M115">View MathML</a>

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M116">View MathML</a>, and the roots<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72">View MathML</a>of Equation (2.10) are repeated; that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73">View MathML</a>;

(c′)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M120">View MathML</a>

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M121">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M77">View MathML</a>are the conjugate pair roots of Equation (2.10).

Proof The similarity between the forms of solutions of Equation (2.1) and solutions of a linear equation with constant coefficients of Equation (2.6) is not just a coincidence.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M124">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M125">View MathML</a>

(2.8)

So that the given equation (2.6) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M126">View MathML</a>

(2.9)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72">View MathML</a> are the two roots of the auxiliary equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M129">View MathML</a>

(2.10)

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M130">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M131">View MathML</a>

The analysis of three cases is similar to Theorem 2.1, we can obtain each solution of the forms as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M132">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M91">View MathML</a>, two distinct real roots,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M134">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M135">View MathML</a>, repeated real roots, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M136">View MathML</a> with the conjugate complex roots.

 □

Remark The constant λ in Equations (2.2) and (2.7) can be solved directly by constant initial value and constant boundary values (or by the numerical methods).

Corollary 2.1The fractional partial differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M137">View MathML</a>

(2.11)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M66">View MathML</a>andA (≠0), B, C, Mare constants, has its solutions of the form given by

(a″)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M141">View MathML</a>

(2.12)

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M142">View MathML</a>;

(b″)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M143">View MathML</a>

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M144">View MathML</a>, and the roots<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72">View MathML</a>of Equation (2.5) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147">View MathML</a>are repeated; that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73">View MathML</a>;

(c″)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M149">View MathML</a>

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M150">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M77">View MathML</a>are the conjugate pair roots of Equation (2.5) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147">View MathML</a>.

Corollary 2.2The fractional partial differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M154">View MathML</a>

(2.13)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M66">View MathML</a>andA (≠0), B, C, Mare constants, has its solutions of the form given by

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M158">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M159">View MathML</a>

(2.14)

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M160">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M161">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M162">View MathML</a>

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M163">View MathML</a>, and the roots<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M72">View MathML</a>of Equation (2.10) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147">View MathML</a>are repeated; that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M73">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M168">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M169">View MathML</a>

when the discriminant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M170">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M77">View MathML</a>are the conjugate pair roots of Equation (2.10) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147">View MathML</a>.

3 Examples

Example 3.1 If the two-dimensional harmonic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M174">View MathML</a> is transformed to plane polar coordinates r and θ, defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M176">View MathML</a>, it takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M177">View MathML</a>

(3.1)

then it has solutions of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M178">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M28">View MathML</a> and λ are constants.

Solution Equation (3.1) is coincident to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M181">View MathML</a>

We have the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M182">View MathML</a>

by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M185">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M186">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M188">View MathML</a> in Theorem 2.1. □

Example 3.2 The fractional partial differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M189">View MathML</a>

Solution Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M193">View MathML</a> in Corollary 2.2, we obtain the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M194">View MathML</a>

where the discriminant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M195">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M196">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M197">View MathML</a>

The analysis of the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M198">View MathML</a> is similar to Theorem 2.2. □

Example 3.3 The fractional partial differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M199">View MathML</a>

Solution Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M183">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M201">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M203">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M204">View MathML</a> in Corollary 2.1, we obtain the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M205">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M206">View MathML</a>

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M207">View MathML</a>.

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M209">View MathML</a>.

If the discriminant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M210">View MathML</a>, the solution is trivial. If the discriminant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M211">View MathML</a>, then the solution is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M212">View MathML</a>

 □

Example 3.4 The fractional partial differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M213">View MathML</a>

Solution Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M214">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M215">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M193">View MathML</a> in Corollary 2.2, the discriminant is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M219">View MathML</a>, but <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M220">View MathML</a> leads to a contradiction, hence there are different real roots <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M222">View MathML</a>, so that we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M223">View MathML</a>

By the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M224">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M225">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M226">View MathML</a>. So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M227">View MathML</a>

and the particular solution is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M228">View MathML</a>

If we apply the Laplace transform to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M229">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M230">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M231">View MathML</a>. Using the residue theorem,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/126/mathml/M232">View MathML</a>

The solution obtained by the method of Laplace transform and the residue theorem is a coincidence, which is our result above. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

SDL carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. CHL carried out the immunoassays. SMS participated in the sequence alignment.

Acknowledgements

The authors are deeply appreciative of the comments and suggestions offered by the referees for improving the quality and rigor of this paper. The present investigation was supported, in part, by the National Science Council of the Republic of China under Grant NSC-101-2115-M-033-002.

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